Problem 23 - Calculus for Biology and Medicine

Question

Use a graphing calculator to construct a \(95 \%\) confidence interval for a sample of size 30 from a uniform distribution over the interval \((0,1)\). Take a class poll to determine the percentage of confidence intervals that contain the true mean. Discuss the result in class.

Step-by-Step Solution

Answer

Construct a 95% confidence interval from a random sample. Determine if it includes the true mean of 0.5. Repeat and discuss with the class.

Step 1: Generate a Random Sample

Use your graphing calculator to generate a random sample of size 30 from a uniform distribution over the interval (0,1). This can often be done using a random number function, such as `rand(30)`, depending on the calculator model.

Step 2: Calculate the Sample Mean

Find the mean of the sample by adding up all the generated numbers and then dividing by 30. Denote this mean by \( \bar{x} \).

Step 3: Calculate the Sample Standard Deviation

Use the standard deviation function on your graphing calculator to find the sample standard deviation, denoted as \( s \). This measures the spread of your data.

Step 4: Determine the Critical t-value

Find the critical t-value for a 95% confidence interval. Since the sample size is 30, with 29 degrees of freedom, use your calculator or a t-table to find the t-value (approximately 2.045 for 95% confidence).

Step 5: Calculate the Margin of Error

Use the formula for the margin of error: \[ E = t \frac{s}{\sqrt{n}} \]where \( t \) is the critical t-value, \( s \) is the sample standard deviation, and \( n \) is the sample size (30).

Step 6: Construct the Confidence Interval

Calculate the confidence interval using the sample mean, margin of error, and the previous confidence interval formula:\[ \text{Confidence Interval} = \left( \bar{x} - E, \bar{x} + E \right) \]

Step 7: Verify if True Mean is Included

The true mean of a uniform distribution over (0,1) is 0.5. Check if the interval from Step 6 includes 0.5.

Step 8: Conduct a Class Poll

Ask each classmate to perform the same process. Record how many of the constructed confidence intervals contain 0.5. Determine the percentage of intervals that contain the true mean.

Step 9: Discuss Results

Review and discuss results as a class. Although the theoretical confidence level is 95%, the actual percentage of intervals containing 0.5 may vary due to random sampling variability.

Key Concepts

Uniform Distribution

A uniform distribution is often described as one in which every outcome has an equal probability of occurring. Imagine you are rolling a fair six-sided die; each outcome from 1 to 6 has an equal chance of 1/6. In this exercise, we are dealing with a continuous uniform distribution over the interval \(0, 1\).
This means that any number between 0 and 1 (including values like 0.23 or 0.87) is equally likely to be chosen when a sample is taken.
Key features of a uniform distribution include:

• Every outcome is equally likely.
• It's characterized by constant probability over its entire range.
• The probability density function is constant.
• The mean, or average, of the distribution is the midpoint of the interval. For \( (0,1) \), the mean is 0.5.

Understanding uniform distribution helps you know that each sample from \(0, 1\) doesn't favor any specific number.

Sample Mean

The sample mean is a crucial statistic that helps summarize the data through a single number. It is the arithmetic average of all the numbers in your data set. Calculating the sample mean involves adding up all the sampled numbers and then dividing by the sample size.
For example, if your sample of size 30 resulted in the numbers 0.1, 0.5, 0.9, and so on, calculate the sum, and then divide by 30 to get the sample mean \( \bar{x} \).
A few important points about the sample mean:

• It's a measure of central tendency, representing the "center" of the data set.
• The sample mean is often used to estimate the population mean.
• If you take new samples, your sample mean might change slightly.

The sample mean provides a snapshot of your data set's typical value.

Critical t-value

The critical t-value is a statistic used in constructing confidence intervals when the sample size is small (typically less than 30) or when the population standard deviation is unknown. For constructing a confidence interval, the critical t-value helps you determine how far from the sample mean the true population mean might be.
This value is found using a t-distribution table or calculator, based on two key things:

• The desired confidence level (like 95%, which is common).
• The number of degrees of freedom, which is one less than the sample size.

Given 30 samples, you have 29 degrees of freedom. For a 95% confidence level, the critical t-value is approximately 2.045.
It helps adjust the confidence interval so it accurately reflects the amount of data you have and the desired certainty level.

Margin of Error

The Margin of Error (MOE) quantifies the uncertainty in estimating the population mean. It describes how much the sample mean might differ from the true population mean.
To calculate the margin of error, use the formula: \[ E = t \frac{s}{\sqrt{n}} \]
Here:

• \( t \) is the critical t-value (approx. 2.045 for 95% confidence and 29 degrees of freedom).
• \( s \) is the sample standard deviation, which measures the spread of the sample data.
• \( n \) is the sample size, for this exercise, it's 30.

The margin of error provides the range in which the true mean might lie, around the sample mean. It's essential for constructing the full confidence interval, allowing us to understand the precision of our sample mean estimate.